Source antenna switching scheme for non-orthogonal protocol

ABSTRACT

The present invention relates to a source antenna switching scheme for a non-orthogonal protocol; and more particularly, to a source antenna switching scheme for a non-orthogonal protocol, which transmits a signal of a source node to a destination node through a relay node. The present invention provides a source antenna switching scheme for a non-orthogonal decode-and-forward protocol that can acquire a greater diversity than the conventional NDF protocol. In other words, the present invention provides a source antenna switching scheme for a non-orthogonal decode-and-forward protocol that can increase a diversity order by adding a reasonable priced antenna instead of expensive hardware such as an RF chain when there are a plurality of antenna in the RF chain.

CROSS-REFERENCE TO RELATED APPLICATION

This application claims priority to Korean Patent Application No.10-2009-0044200 filed on May 20, 2009, the entire contents of which arehereby incorporated by reference.

FIELD OF THE INVENTION

The present invention relates to a source antenna switching scheme for anon-orthogonal protocol; and more particularly, to a source antennaswitching scheme for a non-orthogonal protocol, which transmits a signalof a source node to a destination node through a relay node.

BACKGROUND OF THE INVENTION

Typically, in the radio communication system, a space diversity can beacquired through multiple independent paths between transmitters (sourcenodes) and receivers (destination nodes) that are relative to space-timecodes (STC). Further, if a relay node is used, there are additionalindependent paths between the transmitters and the receivers,penetrating the relay node. Such a system is called a cooperativenetwork. In this case, a cooperative diversity is acquired. In atwo-phase cooperative protocol of the cooperative network, the sourcenode transmits signals to the relay node or the destination node at afirst phase (or timeslot) and the relay node transmits signals to thedestination node at a second phase. The protocol is called anon-orthogonal protocol or an orthogonal protocol according to whetherthe source node continually performs the transmission at the secondphase.

Recently, studies on the cooperative network have been widely conducted.In a paper, J. N. Laneman and G. W. Wornell, “Distributedspace-time-coded protocols for exploiting cooperative diversity inwireless networks,” IEEE Trans. Inform. Theory, vol. 49, pp. 2415˜2425,October 2003, repetition and space-time algorithms has been suggested toacquire the cooperative diversity. Moreover, in a document, Y. Jing andB. Hassibi, “Distributed space-time coding in wireless relay networks,”IEEE Trans. Wireless Commun., vol. 5, no. 12, pp. 3524˜3536, December2006, a distributed STC (DSTC) for an amplify-and-forward (AF) protocolby using a two-hop system has been suggested. If transmission power isinfinitely great, in the scheme, when R relay nodes are used, thediversity order R is acquired. In a paper, Y. Jing and H. Jafarkhani,“Using orthogonal and quasi-orthogonal design in wireless relaynetworks,” IEEE Trans. Inform. Theory, vol. 53, no. 11, pp. 4106˜4118,November 2007, practical DSTCs has been designed by using orthogonalspace-time block codes (OSTBCs) and quasi-orthogonal space-time blockcodes (QOSTBCs) for the AF protocol. In a paper, B. Maham and A.HjÁungnes, “Distributed GABBA space-time codes in amplify-and-forwardcooperation,” in Proc. ITW 2007, July 2007, there has been suggested thedesign of the DSTCs performed by using generalized QOSTBCs. Here, anyrelay nodes can be used to increase the diversity order. Moreover, asuboptimal linear decoder can be used to acquire a maximum diversityorder and reduce the complexity. In a paper, Y. Jing and B. Hassibi,“Cooperative diversity in wireless relay networks with multiple antennanodes,” in Proc. ISIT'05, pp. 815˜819, September 2005, the DSTCs for theAF protocol have expanded to the cooperative network by a multipleantenna. In a paper, G. S. Rajan and B. S. Rajan, “A non-orthogonaldistributed space-time coded protocol—Part 1: Signal model and designcriteria,” in Proc. ITW'06, pp. 385˜389, March 2006, there has beensuggested a non-orthogonal AF (NAF) protocol generalized by using asingle antenna in the source node, the relay node, and object node,respectively. They prove that 3 other protocols have the same diversityorder R+1. Here, a first protocol and a second protocol are identical tothe NAF protocol and the orthogonal AF (OAF) protocol, respectively. Thereason that they have the same diversity order is that signals includingsame information transmitted from the source node undergo same fading atthe first and second phases.

In such aforementioned protocols, the source node signals to the relaynodes and the destination node at the phases. At the first phase, thesource node transmits a STC and the relay node transmits a re-encodedSTC by using a signal decoded from the received signal. However, eventhrough the source node transmits the signal two times at the first andsecond phases. Since a source-destination (SD) channel is the same atthe first and second phases, the DSTCs may not increase the diversityorder.

SUMMARY OF THE INVENTION

In view of the above, the present invention provides a source antennaswitching scheme for a non-orthogonal protocol, which has greaterdiversity than a conventional non-orthogonal protocol.

In accordance with an aspect of the present invention, there is provideda source antenna switching scheme for a non-orthogonal protocol thattransmits a signal of a source node through at least one RF chain havingtwo transmitting antennas, including: selecting any one of the twoantennas of each of the RF chain(s) and allowing a source node totransmit a signal to a relay node and a destination node by using theselected one antenna in a total of M_(S) quantities; and selecting theother of the two antennas and allowing the other antenna in the M_(S)quantities to cooperate with an antenna(s) of the relay node to transmita signal to destination node. The M_(S) may be the number of RF chainsof the source node. The selecting of the other antenna and allowing theother antenna in the M_(S) quantities to cooperate with an antenna(s) ofthe relay node may include allowing the source node and the relay nodeto generate a DSTC and transmit the generated DSTC to the destinationnode. The selecting of the other antenna and allowing the other antennain the M_(S) quantities to cooperate with an antenna(s) of the relaynode may include: decoding or amplifying the received signal by therelay node; and generating a STC by using the decoded or amplifiedsignal to be transmitted by the source node by use of the other antennain the M_(S) quantities and transmitting the generated STC to thedestination node. The selecting of the other antenna and allowing theother antenna in the M_(S) quantities to cooperate with an antenna(s) ofthe relay node may include: decoding the received signal by the relaynode; generating a STC by using the decoded signal to be transmitted bythe source node by use of the other antenna in the M_(S) quantities andtransmitting the generated STC to the destination node; and decoding byusing a near ML decoding method a signal that is received through theselecting of any one of the two antennas of each of the RF chain(s) andallowing a source node to transmit a signal to a relay node and adestination node by using the selected one antenna in a total of M_(S)quantities and the selecting of the other of the two antennas andallowing the other antenna in the M_(S) quantities to cooperate with anantenna(s) of the relay node to transmit a signal to destination node.The near ML decoding method may be performed by a following formula:

${\hat{x} = {\arg{\min\limits_{x \in A^{L}}\{ {{{Y_{D_{1}} - {\sqrt{\frac{p_{1}}{M_{S}}}{{GX}_{1}(x)}}}}^{2} + {\min\limits_{{\hat{x}}_{R} \in A^{L}}\lbrack {{{Y_{D_{2}} - {\sqrt{\frac{p_{2}}{M_{S}}}{{HX}_{2}(x)}} - {\sqrt{\frac{p_{3}}{M_{R}}}{{FX}_{3}( {\hat{x}}_{R} )}}}}^{2} - {\sigma^{2}\ln\;{P_{SR}( xarrow{\hat{x}}_{R} )}}} \rbrack}} \}}}},$in which the G is a channel coefficient matrix of a channel between thesource node and the destination node in the selecting of any one of thetwo antennas of the RF chain(s) and allowing the source node to transmita signal to the relay node and the destination node by using theselected one antenna in the M_(S) quantities, the H is a channelcoefficient matrix of a channel between the source node and thedestination node in the selecting of the other antenna and allowing theother antenna in the M_(S) quantities to cooperate with an antenna(s) ofthe relay node; the F is a channel coefficient matrix of a channelbetween the relay node and the destination node in the selecting of theother antenna and allowing the other antenna in the M_(S) quantities tocooperate with an antenna(s) of the relay node, the x is L data symbolstransmitted from the source node through the selecting of any one of thetwo antennas of the RF chain(s) and allowing the source node to transmita signal to the relay node and the destination node by using theselected one antenna in the quantities and the selecting of the otherantenna and allowing the other antenna in the M_(S) quantities tocooperate with an antenna(s) of the relay node, the M_(R) is the numberof transmitting and receiving antennas of the relay node, the X₁(x) is acode of M_(S)×T₁ of the L data symbols, the X₂(x) is a code of M_(S)×T₂of the L data symbols, the X₃({circumflex over (x)}_(R)) is a code ofM_(R)×T₂ of the L data symbols, the p₁ is a power of a signaltransmitted from the source node in a first operation, the p₂ is a powersupplied from the source node in a second operation, the p₃ is a powersupplied from the relay node in the second operation, the σ² is a powerof noise in the relay node and the destination node, the P_(SR) is apairwise error probability, the S is the source node, the R is the relaynode, and the D is the destination node. The two antennas of each of theRF chain(s) may operate independently from each other.

In accordance with another aspect of the present invention, there isprovided a decoding method of a decode-and-forward protocol thatcooperates with a relay node to transmit a signal of a source node to adestination node through at least one RF chain having two transmittingantennas, including: calculating a metric value for symbol sets{circumflex over (x)}_(R)∈A^(L) that are decodable by the relay node toacquire a smallest value; squaring a distance between a signal receivedby the destination node and a symbol set transmittable by the sourcenode; adding the smallest value acquired by calculating a metric valuefor symbol sets {circumflex over (x)}_(R)∈A^(L) decodable by the relaynode and the squared distance between a signal received by thedestination node and a symbol set transmittable by the source node toacquire and detect a symbol set {circumflex over (x)} having a minimumvalue.

In this case, the calculating of a metric value for symbol sets{circumflex over (x)}_(R)∈A^(L) decodable by the relay node to acquire asmallest value may include: acquiring a pairwise error probability thatan error of a symbol set {circumflex over (x)}_(R) is generated in therelay node when the source node transmits a symbol set x if a signal istransmitted from the source node to the relay node and the destinationnode by selecting any one of the two antennas of each of the RF chain(s)and using the selected one antenna in a total of M_(S) quantities;acquiring a natural log of the pairwise error probability andmultiplying the acquired natural log by a noise power σ² of the relaynode; squaring a distance between the signal received by the destinationnode and symbol sets x and {circumflex over (x)}_(R) transmittable bythe source node and the relay node when a signal is transmitted to thedestination node by allowing the other antenna in the M_(S) quantitiesto cooperate with an antenna(s) of the relay node to transmit a signalto the destination node; and subtracting a value acquired in theacquiring of a natural log of the pairwise error probability andmultiplying the acquired natural log by a noise power σ² of the relaynode from a value acquired in the squaring of a distance between thesignal received by the destination node and symbol sets x and{circumflex over (x)}_(R) transmittable by the source node and the relaynode when a signal is transmitted to the destination node by allowingthe other antenna in the M_(S) quantities to cooperate with anantenna(s) of the relay node to transmit a signal to the destinationnode, and the M_(S) are the number of RF chains of the source node

The calculating of a metric value for symbol sets {circumflex over(x)}_(R)∈A^(L) decodable by the relay node to acquire a smallest valuemay include: acquiring a pairwise error probability that an error of asymbol set {circumflex over (x)}_(R) is generated in the relay node whenthe source node transmits a symbol set x if a signal is transmitted fromthe source node to the relay node and the destination node; acquiring anatural log of the pairwise error probability and multiplying theacquired natural log by a noise power σ² of the relay node; squaring adistance between the signal received by the destination node and symbolsets {circumflex over (x)}_(R) transmittable by the source node and therelay node when a signal is transmitted to the destination node by usingan antenna(s) of the relay node after the source node transmits a signalto the relay node and the destination node; and subtracting a valueacquired in the acquiring of a natural log of the pairwise errorprobability and multiplying the acquired natural log by a noise power σ²of the relay node from a value acquired in the squaring of a distancebetween the signal received by the destination node and symbol sets{circumflex over (x)}_(R) transmittable by the source node and the relaynode when a signal is transmitted to the destination node by using anantenna(s) of the relay node after the source node transmits a signal tothe relay node and the destination node, and the M_(S) are the number ofRF chains of the source node.

BRIEF DESCRIPTION OF THE DRAWINGS

The objects and features of the present invention will become apparentfrom the following description of embodiments, given in conjunction withthe accompanying drawings, in which:

FIG. 1 is a conceptual diagram showing a non-orthogonal protocol inaccordance with an embodiment of the present invention;

FIG. 2 is a flowchart showing a source antenna switching scheme of anon-orthogonal decode-and-forward protocol in accordance with theembodiment of the present invention;

FIG. 3 is a conceptual diagram showing source antenna switchingtransmission of a non-orthogonal protocol in accordance with theembodiment of the present invention;

FIG. 4 is a graph comparing performances of Alamouti scheme of a NDF-SASand a NDF protocol for M_(S)=M_(R)=M_(D)=1, when σ² _(SD)=σ² _(RD)=1, inan error-free SR channel;

FIG. 5 is a graph comparing performances of a NDF-SAS and a NDF protocolfor M_(S)=M_(R)=2 and M_(D)=1, when σ² _(SD)=σ² _(RD)=1, in anerror-free SR channel;

FIG. 6 is a graph comparing performances of an ML decoding method and anear ML decoding method of an ODF protocol using an Alamouti scheme anda QPSK for M_(S)=M_(R)=M_(D)=2, when σ² _(SD)=σ² _(SR)=σ² _(RD)=1, in anerroneous SR channel;

FIG. 7 is a graph comparing performances of Alamouti scheme of a NDF-SASand a NDF protocol for M_(S)=M_(R)=M_(D)=1, when σ² _(SD)=σ² _(RD)=1, inan erroneous SR channel; and

FIG. 8 is a graph comparing performances of a CISTBC scheme of a NDF-SASand a NDF protocol for M_(S)=M_(R)=2 and M_(D)=1, when σ² _(SD)=σ²_(RD)=1, in an erroneous SR channel.

DETAILED DESCRIPTION OF THE EMBODIMENT

Some embodiments of the present invention will be now described indetail with reference to the accompanying drawings which form a parthereof.

However, the present invention is not limited to the embodiments andwill be embodied as various different forms. These embodiments serveonly for making completed the disclosure of the present invention andfor completely informing any person of ordinary skill in the art of thescope of the present invention. Throughout the drawings, similarelements are given similar reference numerals. In the followingmathematical equations, capital letters indicate matrices. I_(n)indicates an n×n unit matrix and ∥•∥ indicates a Frobenius norm definedas the square root of the sum of the absolute squares of its elements.

indicates an expected value.

and

indicate a complex conjugate matrix and a complex conjugate transposematrix, respectively.

indicates a n×m complex matrix. For A∈

^((n×m)), A˜CN(0,σ²I_(nm)) indicates that the elements of A arecircularly symmetric Gaussian random variables of independentlyidentical distribution (i.i.d.) having mean of 0 and variance of σ²Moreover, a non-orthogonal decode-and-forward protocol in accordancewith the embodiment of the present invention is called a NDF-SAS. Ofcourse, the non-orthogonal decode-and-forward protocol is used as anexample in this embodiment, but the present invention is not limited tothis embodiment. The present invention is applicable to allnon-orthogonal protocols performing amplitude as well as decoding.

FIG. 1 is a conceptual diagram showing a non-orthogonal protocol inaccordance with an embodiment of the present invention and FIG. 2 is aflowchart showing a source antenna switching scheme of a non-orthogonaldecode-and-forward protocol in accordance with the embodiment of thepresent invention. FIG. 3 is a conceptual diagram showing source antennaswitching transmission of a non-orthogonal protocol in accordance withthe embodiment of the present invention and FIG. 4 is a graph comparingperformances of Alamouti scheme of the NDF-SAS and the NDF protocols forM_(S)=M_(R)=M_(D)=1, when σ² _(SD)=σ² _(RD)=1, in an error-free SRchannel. FIG. 5 is a graph comparing performances of the NDF-SAS and theNDF protocols for M_(S)=M_(R)=2 and M_(D)=1, when σ² _(SD)=σ² _(RD)=1,in an error-free SR channel. FIG. 6 is a graph comparing performances ofan ML decoding method and a near ML decoding method of an ODF protocolusing an Alamouti scheme and a QPSK for M_(S)=M_(R)=M_(D)=2, when σ²_(SD)=σ² _(SR)=σ² _(RD)=1. Finally, FIG. 7 is a graph comparingperformances of Alamouti scheme of a NDF-SAS and a NDF protocol forM_(S)=M_(R)=M_(D)=1, when σ² _(SD)=σ² _(RD)=1, in an erroneous SRchannel and FIG. 8 is a graph comparing performances of a CISTBC schemeof the NDF-SAS and the NDF protocols for M_(S)=M_(R)=2 and M_(D)=1, whenσ² _(SD)=σ² _(RD)=1, in an erroneous SR channel.

As shown in FIG. 1, a source antenna switching scheme of anon-orthogonal decode-and-forward protocol in accordance with theembodiment of the present invention includes transmitting a signal of asource node to a relay node and a destination node in operation S₁ andgenerating a distributed STC and transmitting the distributed STC to thedestination mode by the source node and the relay node in operation S₂.In this case, the present embodiment assumes half duplex transmissionand a channel is frequency-flat slow fading. It is also assumed that achannel coefficient is not changed from a first stage to a second stageand the destination node is notified of channel state information (CSI)of a source-relay node channel, a source-destination node channel and arelay-destination node channel.

In the operation S₁, a signal of a source node is simultaneouslytransmitted to a relay node and a destination node. The operation S₁includes generating a space-time code in operation S₁₋₁ and transmittingthe space-time code in operation S₁₋₂.

In the operation S₁₋₁, X₁(x) of M_(S) X T₁ from L data symbols X=(x₁,x₂. . . x_(L)) transmitted in the operation S₁.

In this case, a signal matrix received from the relay node and thedestination node can be represented by the following Formula 1 andFormula 2, respectively.

$\begin{matrix}{Y_{R} = {{\sqrt{\frac{p_{1}}{M_{S}}}{{KX}_{1}(x)}} + N_{R}}} & \lbrack {{Formula}\mspace{14mu} 1} \rbrack \\{Y_{D_{1}} = {{\sqrt{\frac{p_{1}}{M_{S}}}{{GX}_{1}(x)}} + N_{D_{1}}}} & \lbrack {{Formula}\mspace{14mu} 2} \rbrack\end{matrix}$

Here, K∈

^(M) ^(R) ^(×M) ^(S) and G∈

^(M) ^(D) ^(×M) ^(S) are channel coefficients between the source nodeand the relay node (SR) and the source node and the destination node(SD) dispersed as CN(0,σ² _(SR)I_(M) _(R) _(M) _(S) ) and CN(0,σ²_(SD)I_(M) _(D) _(M) _(S) ), respectively, in the operation S₁. M_(D) isthe number of receiving antennas in the destination node. M_(R)∈

^(M) ^(R) ^(×T) ¹ and N_(D) ₁ ∈

^(M) ^(D) ^(×T) ¹ are noise matrices distributed as CN(0,σ²I_(M) _(R)_(T) ₁ ) and CN(0,σ²I_(M) _(D) _(T) ₁ ) in the relay node and thedestination node, respectively.

$\rho = \frac{1}{\sigma^{2}}$is a parameter that is linearly in proportion to an average transmissionsignal to noise ratio (SNR), ρ₁ a power of a code sent from the sourcenode in the operation S₁.

In the operation S₁₋₂, the code X₁(x) of M_(S) X T₁ generated in theoperation is transmitted to the relay node and the destination node.

In the operation S₂, a signal that is identical to the signal, i.e., thesignal of the source node transmitted in the operation S₁ isre-transmitted from the source node and the relay node to thedestination node. The re-transmitting of the signal of the source nodeand the signal of the relay node to the destination node includesdecoding the received signal by the relay node in operation S₂₋₁ andgenerating a distributed space-time code with the signal of the sourcenode signal and the signal decoded by the relay node and transmittingthe distributed space-time code in operation S₂₋₂.

In the operation S₂₋₁, the space-time code transmitted from the sourcenode is decoded to generate the space-time code. In this case, asdescribed above, the space-time code can be generated by combining thecodes transmitted from the source node and the relay node, respectively.Such the space-time code (STC) can be represented as the Formula 1.

In the operation S₂₋₂, the distributed space-time code is generated byusing the space-time code generated by combining the codes transmittedfrom the source node and the relay node and the space-time code of thesource node and is transmitted to the destination node.

In this case, when X₂∈

^(M) ^(S) ^(×T) ¹ and X₃∈

^(M) ^(R) ^(×T) ² are code matrices, transmitted from the relay node andthe source node, having transmission powers ρ_(2/M) _(S) and ρ_(3/M)_(R) for each active antenna, M_(R) is the number of receiving andtransmitting antennas in the relay node. Here, in the re-transmitting ofthe signal of the source node and the signal of the relay node to thedestination node, the distributed space-time code (DSTC) can berepresented as the following formula 3 by combining the codestransmitted from the source node and the relay node.

$\begin{matrix}\begin{bmatrix}{\frac{\sqrt{p_{2}}}{M_{S}}{X_{2}(x)}} \\{\frac{\sqrt{p_{3}}}{M_{R}}{X_{3}( x_{R} )}}\end{bmatrix} & \lbrack {{Formula}\mspace{14mu} 3} \rbrack\end{matrix}$

Moreover, the signal received in the destination node can be representedas the following formula 4.

$\begin{matrix}{Y_{D_{2}} = {{\sqrt{\frac{p_{2}}{M_{S}}}{{HX}_{2}(x)}} + {\sqrt{\frac{p_{3}}{M_{R}}}{{FX}_{3}( x_{R} )}} + N_{D_{2}}}} & \lbrack {{Formula}\mspace{14mu} 4} \rbrack\end{matrix}$

Here, H∈

^(M) ^(D) ^(×M) ^(S) and F∈

^(M) ^(D) ^(×M) ^(R) are channel coefficient matrices of channelsbetween the source node and the destination node (SD) and the relay nodeand the destination node (RD), respectively, in a second stage. Theelements of H and P are independently identical distributed (i.i.d.)circularly symmetric complex Gaussian random variables withCN(0,σ²I_(SD)) and CN(0,σ²I_(RD)), respectively. N_(D) ₂ ∈

^(M) ^(D) ^(×T) ² is a noise matrix in the re-transmitting of the signalof the source node and the signal of the relay node to the destinationnode and the element of the noise matrix is an i. i. d. circularlysymmetric complex Gaussian random variables with CN(0,σ²).

Next, an average bit error probability of a conventional NDF protocol iscompared with the source antenna switching scheme of the non-orthogonaldecode-and-forward protocol in accordance with the embodiment of thepresent invention.

Firstly, an error-free channel, i.e., a channel of X_(R)=X will bedescribed to observe the performance of the NDF protocol for aconventional error-free channel. In this case, the equivalentinput-output relation can be simplified as the following formula 5.Y _(D) =H _(e) X _(e)(x)+N _(D)  [Formula 5]

Here, Y_(D)=[Y_(D1) Y_(D2)] is a matrix of the receiving signal andH_(e) is an equivalent channel matrix represented as H_(e)=[G H F].

${X_{e}(x)} = \begin{bmatrix}{\sqrt{\frac{p_{1}}{M_{S}}}{X_{1}(x)}} & 0 \\0 & {\sqrt{\frac{p_{2}}{M_{S}}}{X_{2}(x)}} \\0 & {\sqrt{\frac{p_{3}}{M_{R}}}{X_{3}(x)}}\end{bmatrix}$is an equivalent code matrix and N=[N_(D1) N_(D2)] is a complex Gaussiannoise matrix having CN(0,σ²I_(M) _(D) _((T) ₁ _(+T) ₂ ₎).

There may be induced an average pairwise error probability for a case ofconfusing x with {hacek over (x)} for the equivalent input and outputrelationship of the formula 5. A maximum likelihood (ML) decoding matrixcan be defined as m(Y_(D),X_(e)(x))=∥Y_(D)−H_(e)X_(e)(x)∥². When achannel is known, the pairwise error probability can be represented asthe following formula 6.P(x→{hacek over (x)})=P(m(Y _(D) ,X _(e)({hacek over (x)}))<m(Y _(D) ,X_(e)(x)))  [Formula 6]

Here, m(Y_(D),X_(e)(x))=∥N_(D)∥²

andm(Y _(D) ,{hacek over (X)} _(e))=∥√{square root over (ρ)}H _(e)(X _(e)−{hacek over (X)} _(e))+N _(D)∥² =ρ∥H _(e)(X _(e) −{hacek over (X)}_(e))∥²+2√{square root over (ρ)}Re{tr(H _(e)(X _(e) −{hacek over (X)}_(e))N _(D) ^(†))}+∥N_(D)∥²and Re(•) and tr(•) is a real part of a complex number and a trace of amatrix, respectively. Accordingly, a pairwise error probability can berepresented as the following formula 7.P(x→{hacek over (x)})=P(2Re{tr(H _(e)(X _(e)(x)−X _(e)({hacek over(x)}))N _(D) ^(†))}<−∥H _(e)(X _(e)(x)−X _(e)({hacek over(x)}))∥²)  [Formula 7]

Here, 2Re{tr(H_(e)(X_(e)(x)−X_(e)({hacek over (x)}))N_(D) ^(†))} is areal number Gaussian random variable having a mean of 0 and a varianceof 2σ²∥H_(e)(X_(e)(x)−X_(e)({hacek over (x)}))∥². Accordingly, the apairwise error probability can be represented as the following formula8.

$\begin{matrix}{Q( \sqrt{\frac{1}{2\sigma^{2}}{{H_{e}( {{X_{e}(x)} - {X_{e}( \overset{\Cup}{x} )}} )}}^{2}} )} & \lbrack {{Formula}\mspace{14mu} 8} \rbrack\end{matrix}$

Here, the average pairwise error probability can be represented as thefollowing formula 9 by using

${Q(x)} = {\frac{1}{\sqrt{2\pi}}{\int_{x}^{\infty}{{\mathbb{e}}^{- \frac{\mu^{2}}{2}}{\mathbb{d}y}}}}$as a result of Craig in

$\mspace{79mu}{{Q(x)} = {\frac{1}{\pi}{\int_{x}^{\frac{\pi}{2}}{{\exp\lbrack {- \frac{x^{2}}{2\;\sin^{2}\theta}} \rbrack}{{\mathbb{d}\theta}.\mspace{625mu}\lbrack {{Formula}\mspace{14mu} 9} \rbrack}}}}}$$\begin{matrix}{{E\lbrack {P( xarrow\overset{\_}{x} )} \rbrack} = {E\lbrack {P( xarrow\overset{\_}{x} )} \rbrack}} \\{= {\frac{1}{\pi}{\int_{0}^{\frac{\pi}{2}}{{E\lbrack {\exp\{ {- \frac{{{H_{e}( {{X_{e}(x)} - {X_{e}( \overset{\_}{x} )}} )}}^{2}}{4\sigma^{2}\sin^{2}\theta}} \}} \rbrack}\ {\mathbb{d}\theta}}}}} \\{= {\frac{1}{\pi}{\int_{0}^{\frac{\pi}{2}}( {{E\lbrack {\exp\{ {- \frac{{{\lbrack H_{e} \rbrack_{4}( {{X_{e}(x)} - {X_{e}( \overset{\_}{x} )}} )}}^{2}}{4\sigma^{2}\sin^{2}\theta}} \}} \rbrack}^{M_{D}}\ {\mathbb{d}\theta}} }}} \\{= {\frac{1}{\pi}{\int_{0}^{\frac{\pi}{2}}{( {M_{r}( {{- \frac{1}{4\sigma^{2}\sin}}2\theta} )} )^{M_{D}}{\mathbb{d}\theta}}}}}\end{matrix}$

Here, [H_(e)]_(i) is an i^(th)-order row of H_(e) andΓ=∥[H_(e)]_(i)(X_(e)(x)−X_(e)({hacek over (x)}))∥² andM_(Γ)(s)=E[exp(aΓ)]. Since the rows of H_(e) has the same statisticcharacteristics, the equality is maintained. The average pairwise errorprobability of the following formula 10 can be obtained by manipulatinga moment generating function M_(Γ)(−1/4σ² sin²θ)) by using of thefollowing formula 20.

$\begin{matrix}{{E\lbrack {P( xarrow\overset{\_}{x} )} \rbrack} = {\frac{1}{\pi}{\int_{0}^{\frac{\pi}{2}}{{{I + {\frac{1}{4\sigma^{2}\sin^{2}\theta}{E\lbrack {\lbrack H_{e} \rbrack_{4}^{\dagger}\lbrack H_{e} \rbrack}_{4} \rbrack}( {{X_{e}(x)} - \mspace{290mu}{X_{e}( \overset{\_}{x} )}} )( {{X_{e}(x)} - {X_{e}( \overset{\_}{x} )}} )^{\dagger}}}}^{- M_{D}}{{\mathbb{d}\theta}.}}}}} & \lbrack {{Formula}\mspace{14mu} 10} \rbrack\end{matrix}$

Here, for the high signal to noise ratio (SNR), if a difference matrixX_(e)(x)−X_(e)({hacek over (x)}) is a full rank, the rank ofE[[H_(e)]_(i) ^(†)[H_(e)]_(i)] determines a diversity of the averagepairwise error probability. Accordingly, by using the source antennaswitching scheme of the non-orthogonal decode-and-forward protocol(hereinafter, referred to as “NDF-ASA protocol”) in accordance with theembodiment of the present invention, it is possible to increase thediversity by M_(S)M_(D) as compared with the conventional NDF protocol.

Next, a plurality of schemes of the NDF-SAS protocol whenM_(S)=M_(R)=M_(D)=1, M_(S)=M_(R)=2, and M_(D)=1 will be described inaccordance with the embodiment of the present invention.

An Alamouti code acquires a maximum speed and a maximum diversity fortwo transmitting antennas. A QOSTBC and a CISTBC having constellationrotation (QOSTBC-CR) acquire a maximum speed and a maximum diversity forfour transmitting antennas. Accordingly, a plurality of code designs forthe NDF-SAS protocol will be described below by using the alamouti code,the CISTBC, and the QOSTBC-CR.

When it is assumed that

${{A( {x_{1},x_{2}} )} = \begin{bmatrix}x_{1} & {- x_{2}^{*}} \\x_{2} & x_{1}^{*}\end{bmatrix}},$for M_(S)=M_(R)=M_(D)=1, a formula X₁(x)=[x₁,x₂] is used in an operationof transmitting the signal of the source node to the relay node and thedestination node (hereinafter, referred to as “a first operation”), andAlamouti schemes X₂(x)=[x₁−x*₂] and X₃(x)=[x₂ x*₁] are used in anoperation of re-transmitting the signal of the source node and thesignal of the relay node to the destination node (hereinafter, referredto as “a second operation).

In the first operation, a formula X₁(x)=[A(x₁,x₂) A(x₃x₄)] is used forM_(S)=M_(R)=2 and M_(D)=1. In the second operation, CISTBC schemesX₂(x)=[A({tilde over (s)}₁,{tilde over (s)}₂) 0] and X₃(x)=[0 A({tildeover (s)}₃,{tilde over (s)}₄)] QOSTBC-CR schemes X₂(x)=[A(x₁,x₂)A(s₃,s₄)] and X₃(x)=[A(s₃,s₄)] are used. Here, s_(i)=x_(i)e^(jθ) and θis a rotation angle. {tilde over (s)}_(i)=s_(i,R)+js_(((i+1)mod 4)+1,I);i=1,2,3,4 and s_(i,R) and s_(i,I) are a real part and a complex part,respectively, of s_(i).

In accordance with the embodiment of the present invention, it isassumed that ρ₁+ρ₂+ρ₃=2 in the alamouti scheme. Accordingly, ρ is anaverage transmission signal to noise ratio (SNR). (X_(e)(x)−X_(e)({hacekover (x)}))(X_(e)(x)−X_(e)({hacek over (x)}))^(†)=(|x₁−{hacek over(x)}₁|²+|x₂−{hacek over (x)}₂|²)diag(ρ₁ρ₂ρ₃). Here, diag(•) is adiagonal matrix. This code may be a decoding single symbol. Moreover,when X_(e)(x_(k)) is assumed to be X_(e) by considering x_(i)=0 for allcases of i≠k, the average pairwise error probability. The pairwise errorprobability of the following formula 11 is acquired from the formula 10.

$\begin{matrix}{{E\lbrack {P( x_{k}arrow\overset{\_}{x_{k}} )} \rbrack} = {\frac{1}{\pi}{\int_{0}^{\frac{\pi}{2}}{{{I + {\frac{1}{4\sigma^{2}\sin^{2}\theta}{E\lbrack {H_{e}^{\dagger}H_{e}} \rbrack}( {{X_{e}( x_{k} )} - \mspace{290mu}{X_{e}( \overset{\_}{x_{k}} )}} )( {{X_{e}( x_{k} )} - {X_{e}( \overset{\_}{x_{k}} )}} )^{\dagger}}}}^{- 1}{\mathbb{d}\theta}}}}} & \lbrack {{Formula}\mspace{14mu} 11} \rbrack\end{matrix}$

Here, E[H_(e) ^(†)H_(e)]=diag(σ² _(SD),σ² _(SD),σ² _(RD)) for theNDF-SAS protocol.

${B\lbrack {H_{e}^{\dagger}H_{e}} \rbrack} = {\begin{matrix}\sigma_{SD}^{2} & \sigma_{SD}^{2} & 0 \\\sigma_{SD}^{2} & \sigma_{SD}^{2} & 0 \\0 & 0 & \sigma_{RD}^{2}\end{matrix}}$for the NDF protocol. Accordingly, the average pairwise errorprobabilities for the NDF-SAS protocol and the NDF protocol,respectively, can be acquired as the following formulae 12 and 13.

$\begin{matrix}{{B\lbrack {P( x_{k}arrow{\overset{\bigvee}{x}}_{k} )} \rbrack} = {\frac{1}{\pi}{\int_{0}^{\frac{\pi}{2}}{( {1 + \frac{p_{1}\sigma_{SD}^{2}\Delta_{x_{k}}^{2}}{4\sigma^{2}\sin^{2}\theta}} )^{- 1}( {1 + \frac{p_{2}\sigma_{SD}^{2}\Delta_{x_{k}}^{2}}{4\sigma^{2}\sin^{2}\theta}} )^{- 1}( {1 + \frac{p_{3}\sigma_{SD}^{2}\Delta_{x_{k}}^{2}}{4\sigma^{2}\sin^{2}\theta}} )^{- 1}\ {\mathbb{d}\theta}}}}} & \lbrack {{Formula}\mspace{14mu} 12} \rbrack \\{{B\lbrack {P( x_{k}arrow{\overset{\bigvee}{x}}_{k} )} \rbrack} = {\frac{1}{\pi}{\int_{0}^{\frac{\pi}{2}}{( {1 + \frac{( {p_{1} + p_{2}} )\sigma_{SD}^{2}\Delta_{x_{k}}^{2}}{4\sigma^{2}\sin^{2}\theta}} )^{- 1}( {1 + \frac{p_{3}\sigma_{RD}^{2}\Delta_{x_{k}}^{2}}{4\sigma^{2}\sin^{2}\theta}} )^{- 1}{\mathbb{d}\theta}}}}} & \lbrack {{Formula}\mspace{14mu} 13} \rbrack\end{matrix}$

Here, Δ_(x) _(k) =|x_(k)−{hacek over (x)}_(k)|,k=1,2. The diversityorder three is acquired from the formulae 12 and 13 by the NDF-SASprotocol. This means that the NDF-SAS protocol is grater than the NDFprotocol of the diversity order two.

In the embodiment of the present invention, it is assumed that2p1+p2+p3=4 for the CISTBC. Accordingly, ρ=1/σ² is an averagetransmission signal to noise ratio (SNR). In this case, since

${( {{X_{e}(x)} - {X_{e}( \overset{\bigvee}{x} )}} )( {{X_{e}(x)} - {X_{e}( \overset{\bigvee}{x} )}} )^{\dagger}} = {\sum\limits_{k = 1}^{4}\;{( {{X_{e}( x_{k} )} - {X_{e}( {\overset{\bigvee}{x}}_{k} )}} )( {{{X_{e}( x_{k} )} - {X_{e}( {\overset{\bigvee}{x}}_{k} )}},} }}$the average pairwise error probability can be independently induced andthe formula 11 can be used to induce the average pairwise errorprobability. When it is assumed that Δ_(s) _(k,R) =|s_(k,R)−{hacek over(s)}_(k,R)| and Δ_(s) _(k,I) =|s_(k,I)−{hacek over (s)}_(k,I)|,(X_(e)(x_(k))−X_(e)({hacek over (x)}_(k)))(X_(e)(x_(k))−X_(e)({hacekover (x)}_(k)))^(†) can be represented as the following formula 14.

             [Formula  14]${( {{X_{e}( x_{k} )} - {X_{e}( ( \overset{\Cup}{x} )_{k} )}} )( {{X_{e}( x_{k} )} - {X_{e}( {\overset{\bigvee}{x}}_{k} )}} )} = \{ \begin{matrix}{{\frac{1}{2}{{diag}( {{p_{1}\Delta_{x_{k}}^{2}},{p_{1}\Delta_{x_{k}}^{2}},{p_{2}\Delta_{x_{k,R}}^{2}},{p_{2}\Delta_{x_{k,R}}^{2}},{p_{3}\Delta_{x_{k,I}}^{2}p_{3}\Delta_{x_{k,I}}^{2}}} )}},} & {{{{for}\mspace{14mu} k} = 1},2} \\{{\frac{1}{2}{{diag}( {{p_{1}\Delta_{x_{k}}^{2}},{p_{1}\Delta_{x_{k}}^{2}},{p_{2}\Delta_{x_{k,I}}^{2}},{p_{2}\Delta_{x_{k,I}}^{2}},{p_{3}\Delta_{x_{k,R}}^{2}p_{3}\Delta_{x_{k,R}}^{2}}} )}},} & {{{{for}\mspace{14mu} k} = 3},4}\end{matrix} $

Moreover, the following formula 15 is for the NDF-SAS protocol and thefollowing formula 16 is for the NDF protocol.

$\begin{matrix}{{B\lbrack {H_{e}^{\dagger}H_{e}} \rbrack} = {{diag}( {\sigma_{SD}^{2},\sigma_{SD}^{2},\sigma_{SD}^{2},\sigma_{SD}^{2},\sigma_{RD}^{2},\sigma_{RD}^{2}} )}} & \lbrack {{Formula}\mspace{14mu} 15} \rbrack \\{{B\lbrack {H_{e}^{\dagger}H_{e}} \rbrack} = {\begin{matrix}{\sigma_{SD}^{2}I_{2}} & {\sigma_{SD}^{2}I_{2}} & 0 \\{\sigma_{SD}^{2}I_{2}} & {\sigma_{SD}^{2}I_{2}} & 0 \\0 & 0 & {\sigma_{RD}^{2}I_{2}}\end{matrix}}} & \lbrack {{Formula}\mspace{14mu} 16} \rbrack\end{matrix}$

If the formulae 15 and 16 are put to the formula 11, the averagepairwise error probability for the NDF-SAS protocol can be representedas the following formula 17.

             [Formula  17]${E{{P( x_{k}arrow{\overset{\bigvee}{x}}_{k} )}}} = \{ \begin{matrix}{{\frac{1}{\pi}{\int_{0}^{\frac{\pi}{2}}{( {1 + \frac{p_{1}\sigma_{SD}^{2}\Delta_{x_{k}}^{2}}{8\sigma^{2}\sin^{2}\theta}} )^{- 2}( {1 + \frac{p_{2}\sigma_{SD}^{2}\Delta_{x_{k,R}}^{2}}{8\sigma^{2}\sin^{2}\theta}} )^{- 2}( {1 + \frac{p_{3}\sigma_{RD}^{2}\Delta_{x_{k,I}}^{2}}{8\sigma^{2}\sin^{2}\theta}} )^{- 2}\ {\mathbb{d}\theta}}}},} \\{{{{for}\mspace{14mu} k} = 1},2} \\{{\frac{1}{\pi}{\int_{0}^{\frac{\pi}{2}}{( {1 + \frac{p_{1}\sigma_{SD}^{2}\Delta_{x_{k}}^{2}}{8\sigma^{2}\sin^{2}\theta}} )^{- 2}( {1 + \frac{p_{2}\sigma_{SD}^{2}\Delta_{x_{k,I}}^{2}}{8\sigma^{2}\sin^{2}\theta}} )^{- 2}( {1 + \frac{p_{3}\sigma_{RD}^{2}\Delta_{x_{k,R}}^{2}}{8\sigma^{2}\sin^{2}\theta}} )^{- 2}\ {\mathbb{d}\theta}}}},} \\{{{{for}\mspace{14mu} k} = 3},4}\end{matrix} $

Since Δ_(x) _(k) ²≠0, Δ_(s) _(k,R) ²≠0, Δ_(s) _(k,I) ²≠0 when σ²→0, thediversity order becomes six. Similarly, the formulae 14 and 16 are putto the formula 11, the average pairwise error probability for the NDFprotocol can be represented as the following formula 18.

$\begin{matrix}{\mspace{625mu}{\lbrack {{Formula}\mspace{14mu} 18} \rbrack{{E\lbrack {P( x_{k}arrow{\overset{\bigvee}{x}}_{k} )} \rbrack} = \{ \begin{matrix}{{\frac{1}{\pi}{\int_{0}^{\frac{\pi}{2}}{( {1 + \frac{\sigma_{SD}^{2}( {{p_{1}\Delta_{x_{k}}^{2}} + {p_{2}\Delta_{x_{k,R}}^{2}}} )}{8\sigma^{2}\sin^{2}\theta}} )^{- 2}( {1 + \frac{\sigma_{RD}^{2}p_{3}\Delta_{x_{k,I}}^{2}}{8\sigma^{2}\sin^{2}\theta}} )^{- 2}{\mathbb{d}\theta}}}},} \\{{{{for}\mspace{14mu} k} = 1},2} \\{{\frac{1}{\pi}{\int_{0}^{\frac{\pi}{2}}{( {1 + \frac{\sigma_{SD}^{2}( {{p_{1}\Delta_{x_{k}}^{2}} + {p_{2}\Delta_{x_{k,I}}^{2}}} )}{8\sigma^{2}\sin^{2}\theta}} )^{- 2}( {1 + \frac{\sigma_{RD}^{2}p_{3}\Delta_{x_{k,R}}^{2}}{8\sigma^{2}\sin^{2}\theta}} )^{- 2}{\mathbb{d}\theta}}}},} \\{{{{for}\mspace{14mu} k} = 3},4.}\end{matrix} }}} & \;\end{matrix}$

In other words, the diversity order of the average pairwise errorprobability for the NDF protocol is four. If it is assumed thatρ₁+ρ₂+ρ₃=2 for the QOSTBC-CR scheme, ρ is the signal to noise ratio(SNR).

Since (X_(e)(x)−X_(e)({hacek over (x)}))(X_(e)(x)−X_(e)({hacek over(x)}))^(†)=(X_(e)(x₁,x₃)−X_(e)({hacek over (x)}₁,{hacek over(x)}₃))(X_(e)(x₁,x₃)−X_(e)({hacek over (x)}₁,{hacek over(x)}₃))^(†)+(X_(e)(x₂,x₄)−X_(e)({hacek over (x)}₂,{hacek over(x)}₄))(X_(e)(x₂,x₄)−X_(e)({hacek over (x)}₂,{hacek over (x)}₄))^(†)(when X_(e)(x_(k)) is assumed to be X_(e) by considering x_(i)=0 for allcases of i≠k), the average pairwise error probability can be induced ina pair for (x₁,x₃) and (x₂,x₄).

In the embodiment of the present invention, X_(e)(x₁,x₃) is consideredto induce the average pairwise error probability and X(x)=X_(e)(x₁,x₃)and X({hacek over (x)})=X_(e)({hacek over (x)}₁,{hacek over (x)}₃).

From the above average pairwise error probability, the following formula19 can be acquired.

$\begin{matrix}{{{E\lbrack {P( xarrow\overset{\bigvee}{x} )} \rbrack} = {\frac{1}{\pi}{\int_{0}^{\frac{\pi}{2}}{{{I + {\frac{1}{4\sigma^{2}\sin^{2}\theta}{E\lbrack {H_{e}^{\dagger}H_{e}} \rbrack}( {{X(x)} - {X( \overset{\bigvee}{x} )}} )( {{X(x)} - {X( \overset{\bigvee}{x} )}} )^{\dagger}}}}^{- 1}{\mathbb{d}\theta}}}}}\mspace{79mu}{{Here},{{( {{X(x)} - {X( \overset{\bigvee}{x} )}} )( {{X(x)} - {X( \overset{\bigvee}{x} )}} )^{\dagger}} = {\frac{1}{2}\begin{bmatrix}{c_{1}p_{1}I_{2}} & 0 & 0 \\0 & {c_{2}p_{2}I_{2}} & {d\sqrt{p_{2}p_{3}}I_{2}} \\0 & {d\sqrt{p_{2}p_{3}}I_{2}} & {c_{2}p_{3}I_{2}}\end{bmatrix}}},}} & \lbrack {{Formula}\mspace{14mu} 19} \rbrack\end{matrix}$c₁=|x₁−{circumflex over (x)}₁|²+|x₃−{hacek over (x)}₃|², c₂=|x₁−{hacekover (x)}₁|²+|s₃−{hacek over (s)}₃|² and d=2Re{(x₁−{hacek over(x)}₁)(s₃−{hacek over (s)}₃)*}. Since s_(k)=x_(k)e^(jΦ), c₁=c₂.

Accordingly, the average pairwise error probability for the NDF-SASprotocol can be represented as the following formula 20.

$\begin{matrix}{{{E\lbrack {P( xarrow\overset{\bigvee}{x} )} \rbrack} = {\frac{1}{\pi}{\int_{0}^{\frac{\pi}{2}}{( {1 + \frac{c_{2}p_{1}\sigma_{SD}^{2}}{8\sigma^{2}\sin^{2}\theta}} )^{- 2}( {1 + \frac{c_{2}p_{2}\sigma_{SD}^{2}}{8\sigma^{2}\sin^{2}\theta} + \frac{c_{2}p_{3}\sigma_{RD}^{2}}{8\sigma^{2}\sin^{2}\theta} + {\frac{p_{2}p_{3}\sigma_{SD}^{2}\sigma_{RD}^{2}}{64\sigma^{4}\sin^{4}\theta}( {c_{2}^{2} - d^{2}} )}} )^{- 2}{\mathbb{d}\theta}}}}},} & \lbrack {{Formula}\mspace{14mu} 20} \rbrack\end{matrix}$

Here, c₂ ²−d²>0 and the diversity order is six. The average pairwiseerror probability for the NDF protocol can be represented as thefollowing formula 21.

$\begin{matrix}{{E\lbrack {P( xarrow\overset{\bigvee}{x} )} \rbrack} = \mspace{76mu}{\frac{1}{\pi}{\int_{0}^{\frac{\pi}{2}}{( {1 + \frac{{c_{2}( {p_{1} + p_{2}} )}\sigma_{SD}^{2}}{8\sigma^{2}\sin^{2}\theta} + \frac{c_{2}p_{3}\sigma_{RD}^{2}}{8\sigma^{2}\sin^{2}\theta} + \mspace{124mu}\frac{c_{2}^{2}p_{1}p_{3}\sigma_{SD}^{2}\sigma_{RD}^{2}}{64\sigma^{4}\sin^{4}\theta} + {\frac{p_{2}p_{3}\sigma_{SD}^{2}\sigma_{RD}^{2}}{64\sigma^{4}\sin^{4}\theta}( {c_{2}^{2} - d^{2}} )}} )^{- 2}{{\mathbb{d}\theta}.}}}}} & \lbrack {{Formula}\mspace{14mu} 21} \rbrack\end{matrix}$

For the NDF protocol, the diversity order is six.

Table 1 shows the diversity orders of the NDF-SAS and NDF protocolsusing various DSTCs for M_(S)=M_(R)=M_(D)=1, M_(S)=M_(R)=2, and M_(D)=1

TABLE 1 NDF-SAS NDF M_(S) = M_(R) = M_(D) = 1_((Alamouti)) 3 2 M_(S) =M_(R) = 2, M_(D) = 1 CISTBC 6 4 QOSTBC

As shown in Table 1, the diversity order of the NDF-SAS in accordancewith the embodiment of the present invention is higher than that of theconventional NDF. Below described is a simulation for checking thediversity orders of the Table 1.

For the simulation, the whole transmission power is the same in thefirst stage and second stage. The power is evenly allotted to eachtransmitting antenna. Accordingly, when M_(S)=M_(R)=M_(D)=1, it isassumed that P₁=1, P₂=0.5, and P₃=0.5 for the NDF-SAS protocol and theNDF protocol.

When M_(S)=M_(R)=2 and M_(D)=1, it is assumed that P₁=1, P₂=1, and P₃=1for the CISTBC scheme and P₁=1,

${p_{2} = \frac{1}{2}},\mspace{14mu}{{{and}\mspace{14mu} p_{3}} = \frac{1}{2}}$for the QOSTBC-CR scheme to make the whole transmitting even. For theQPSK modification, an optimized rotating angle is −31.7175° for theCISTBC scheme and −45° for the QOSTBC-CR scheme in accordance with theembodiment of the present invention.

To compare the average pairwise error probabilities of the NDF-SAS andNDF protocols, FIGS. 4 and 5 show simulation results of σ² _(SD)=σ²_(RD)=1 for M_(S)=M_(R)'M_(D)=1, and M_(S)=M_(R)=2 and M_(D)=1,respectively. It is recognized that the diversity order of the NDF-SASis higher than that of the conventional NDF from the FIGS. 4 and 5.According to the analysis and calculating result, it is recognized thatthe diversity order of the NDF-SAS protocol is improved for theerror-free SR channel.

Next, a performance of the NDF protocol for a conventional error SRchannel will be described.

Firstly, a maximum likelihood (ML) decoding scheme of the NDF protocolfor the error SR channel is defined. In this case, since it isimpossible to use a maximum cooperation between the source node and therelay node, the signals transmitted from the relay node and the sourcenode, respectively, are not accurately identical to each other.Accordingly, a maximum likelihood decoder (hereinafter, referred to as“an ML decoder”) can be represented as the following formula 22.

$\begin{matrix}\begin{matrix}{\hat{x} = {\arg{\max\limits_{x \in A^{L}}{p( {{Y_{{D\; 1};}Y_{D\; 2}}❘x} )}}}} \\{= {\arg{\max\limits_{x \in A^{L}}{\sum\limits_{{\hat{x}}_{R} \in A^{L}}{{p( {Y_{D\; 1},{Y_{D\; 2}❘x},{\hat{x}}_{R}} )}{P_{SR}( {{\hat{x}}_{R}❘x} )}}}}}} \\{= {\arg{\max\limits_{x \in A^{L}}{{p( {Y_{D\; 1}❘{X_{1}(x)}} )}{\sum\limits_{{\hat{x}}_{R} \in A^{L}}{p\begin{matrix}( {{Y_{D\; 2}❘{X_{2}(x)}},{X_{3}( {\hat{x}}_{R} )}} ) \\{P_{SR}( {{\hat{x}}_{R}❘x} )}\end{matrix}}}}}}} \\{= {\arg{\max\limits_{x \in A^{L}}\lbrack {{- \frac{{{Y_{D\; 1} - {\sqrt{\frac{p_{1}}{M_{S}}}{{GX}_{1}(x)}}}}^{2}}{\sigma^{2}}} + {\ln{\sum\limits_{{\hat{x}}_{R} \in A^{L}}{\exp( \frac{{- {{Y_{D\; 2} - {\sqrt{\frac{p_{2}}{M_{S}}}{{HX}_{2}(x)}} - {\sqrt{\frac{p_{3}}{M_{R}}}{{FX}_{3}( {\hat{x}}_{R} )}}}}^{2}} + {\sigma^{2}\ln\;{P_{SR}( {{\hat{x}}_{R}❘x} )}}}{\sigma^{2}} )}}}} \rbrack}}}\end{matrix} & \lbrack {{Formula}\mspace{14mu} 22} \rbrack\end{matrix}$

Here, A is a signal set for a M-ary signal constellation andP_(SR)({circumflex over (x)}_(R)|x) is a probability that the relay nodedecodes a received signal for {circumflex over (x)}_(R) when the sourcenode transmits x in the first stage.

In the formula 22, since it is difficult to P_(SR)({circumflex over(x)}_(R)|x) induce for a code x₁, P_(SR)(x→{circumflex over (x)}_(R)) isused in accordance with the embodiment of the present invention,P_(SR)(x→{circumflex over (x)}_(R)) is a PEP determining {circumflexover (x)}_(R) in the relay node when x is transmitted from the sourcenode. Even if the pairwise error probability is not identical toP_(SR)({circumflex over (x)}_(R)|x), P_(SR)({circumflex over (x)}_(R)|x)can be used to acquire a solution of the formula 22. Further, anapproximate value of

${\sum\limits_{i}\;{\mathbb{e}}^{x_{i}}} \approx {\max_{i}x_{i}}$is used for a high signal to noise ratio (SNR), i.e., σ²→0. Eventually,the ML decoder approaches to the following formula 23.

$\begin{matrix}{\hat{x} = {\arg{\min\limits_{x \in A^{L}}\{ {{{Y_{D\; 1} - {\sqrt{\frac{p_{1}}{M_{S}}}{{GX}_{1}(x)}}}}^{2} + \mspace{40mu}{\min\limits_{{\hat{x}}_{R} \in A^{L}}\lbrack {{{Y_{D\; 2} - {\sqrt{\frac{p_{2}}{M_{S}}}{{HX}_{2}(x)}} - {\sqrt{\frac{p_{3}}{M_{R}}}{{FX}_{3}( {\hat{x}}_{R} )}}}}^{2} - \mspace{436mu}{\sigma^{2}\ln\;{P_{SR}( xarrow{\hat{x}}_{R} )}}} \rbrack}} \}}}} & \lbrack {{Formula}\mspace{14mu} 23} \rbrack\end{matrix}$

In this case, the decoder is called a near ML decoder. Of course, thepresent embodiment applies the near ML decoder to a non-orthogonaldecode-and-forward protocol. The present invention is not limited to theembodiment. In accordance with the present invention, the near MLdecoder can be applied to a orthogonal decode-and-forward protocol inwhich the source node transmits a signal to the relay node and thedestination node in the first stage and only the relay node transmits asignal to the destination node in the second stage.

In other words, in accordance with the present invention, thenon-orthogonal decode-and-forward protocol acquires a smallest value bycalculating a matrix value for symbol sets {circumflex over(x)}_(R)∈A^(L) decodable by the relay node and squares a distancebetween the signal received by the destination node and of the symbolsets x transmittable by the source node. Further, the non-orthogonaldecode-and-forward protocol adds the smallest value acquired bycalculating a matrix value for symbol sets {circumflex over(x)}_(R)∈A^(L) decodable by the relay node and a value obtained bysquaring the distance between the signal received by the destinationnode and the symbol sets x transmittable by the source node. Then, adecoding process of the destination node is completed by detecting asymbol set {circumflex over (x)} having a minimum value acquired byadding the smallest value acquired by calculating a matrix value forsymbol sets {circumflex over (x)}_(R)∈A^(L) decodable by the relay nodeand a value obtained by squaring the distance between the signalreceived by the destination node and the symbol sets x transmittable bythe source node, for all transmittable symbol sets x.

In this case, the calculation of a matrix value is performed as follows.In a first stage where the source node transmits a signal to the relaynode and the destination node, when the source node transmits a symbolset x, a pairwise error probability in which the relay node generates anerror of the symbol set {circumflex over (x)}_(R). Then, in a secondstage where a natural log of the pairwise error probability is acquired,a noise power σ² of the relay node is multiplied, and the source nodeand the relay node cooperate to transmit a signal to the destinationnode, a distance between the signal received by the destination node andsymbol sets x and {circumflex over (x)}_(R) transmittable by the sourcenode and the relay node are squared. Finally, in the second stage, thecalculation of a matrix value can be performed by subtracting a valueacquired by squaring the distance between the signal received by thedestination node and symbol sets x and {circumflex over (x)}_(R)transmittable by the source node and the relay node and another valueacquired by obtaining the natural log of the pairwise error probabilityand by multiplying the natural log by a noise power σ² of the relaynode.

If such a near ML decoding method is applied to the orthogonaldecode-and-forward protocol, the calculation of a matrix value can beperformed by blocking the signal transmission of the source node in thesecond stage. In other words, while calculating the matrix value of theaforementioned non-orthogonal decode-and-forward protocol, only therelay node is allowed to transmit the signal to the destination node toapply the near ML decoding method to the orthogonal decode-and-forwardprotocol. See FIG. 6 for the orthogonal decode-and-forward protocol. Asshown as the near ML decoding method has the same performance as the MLdecoding method.

Next, in accordance with the embodiment of the present invention, anupper limit of the pairwise error probability is induced. Since therelay node can transmit any symbol, the average pairwise errorprobability can be represented as the following formula 24.

$\begin{matrix}{{E\lbrack {P( xarrow\overset{\bigvee}{x} )} \rbrack} = {\sum\limits_{{\hat{x}}_{R} \in A^{L}}{E\lbrack {{P( { xarrow\overset{\bigvee}{x} ❘x_{R}} )}{P_{SR}( {x_{R}❘x} )}} \rbrack}}} & \lbrack {{Formula}\mspace{14mu} 24} \rbrack\end{matrix}$

Here, P(x→{hacek over (x)}|x_(R)) is a conditional pairwise errorprobability that the destination node determines {hacek over (x)} when xand x_(R) are transmitted from the source node and the relay node,respectively. It is assumed that the source node transmits a signal x na first stage and the relay node decodes the received signal for x_(R)to transmit the decoded signal to the destination node in a secondstage. Then, the transmission pairwise error probability of the formula24 can be represented as the following formula 25.P(x→{hacek over (x)}|x _(R))=P(m([Y _(D) ₁ ,Y _(D) ₂ ],{hacek over(x)}|x,x _(R))<m([Y _(D) ₁ ,Y _(D) ₂ ],x|x,x _(R)))  [Formula 25]

Here, m([Y_(D) ₁ ,Y_(D) ₂ ],x|x,x_(R)) and m([Y_(D) ₁ ,Y_(D) ₂ ],{hacekover (x)}|x,x_(R)) are matrices of the formula 23 that determine x and{hacek over (x)} for x and x_(R) transmitted from the source node andthe relay node.

P_(SR)=(x→{hacek over (x)}_(R)) for a SR channel can be represented asthe following formula 26 by analysis of the aforementioned averagepairwise error probability.

$\begin{matrix}{{{P_{SR}( xarrow{\hat{x}}_{R} )} = {Q( \sqrt{\frac{p_{1}}{2\sigma^{2}M_{S}}{{K( {{X_{1}(x)} - {X_{1}( {\hat{x}}_{R} )}} )}}^{2}} )}}{{Here},\mspace{14mu}{{\lim\limits_{\sigma^{2}arrow 0}\;{\sigma^{2}\ln\;{P_{SR}( xarrow z )}}} = {{0\mspace{14mu}{for}\mspace{14mu} z} = {x.\;{Otherwise}}}},\begin{matrix}{{\lim\limits_{\sigma^{2}arrow 0}\;{\sigma^{2}\ln\;{P_{SR}( xarrow z )}}} = {\lim\limits_{\sigma^{2}arrow 0}\;{\sigma^{2}\ln\;{Q( \sqrt{\frac{p_{1}}{2\sigma^{2}M_{S}}{{K( {{X_{1}(x)} - {X_{1}( {\hat{x}}_{R} )}} )}}^{2}} )}}}} \\{= {\lim\limits_{\sigma^{2}arrow 0}\;{\sigma^{2}\ln\frac{{\mathbb{e}}^{- \frac{p_{1}{{K{({{X_{1}{(x)}} - {X_{1}{({\hat{x}}_{R})}}})}}}^{2}}{4\sigma^{2}M_{S}}}}{\sqrt{2\pi\frac{p_{1}}{2\sigma^{2}M_{S}}{{K( {{X_{1}(x)} - {X_{1}( {\hat{x}}_{R} )}} )}}^{2}}}}}} \\{= {{- \frac{p_{1}}{4\; M_{s}}}{{K( {{X_{1}(x)} - {X_{1}( {\hat{x}}_{R} )}} )}}^{2}}}\end{matrix}}} & \lbrack {{Formula}\mspace{14mu} 26} \rbrack\end{matrix}$

Accordingly, for the high signal to noise ratio (SNR), the matrix of theformula 25 approaches to the following formulae 27 and 28.

$\begin{matrix}{{m( {\lbrack {Y_{D_{1}},Y_{D_{2}}} \rbrack,{x❘x},x_{R}} )} \approx {{N_{D_{1}}}^{2} + {\min\limits_{{\hat{x}}_{R} \in A^{L}}\lbrack {{{{\sqrt{\frac{p_{3}}{M_{R}}}{F( {{X_{3}( x_{R} )} - {X_{3}( {\hat{x}}_{R} )}} )}} + N_{D_{2}}}}^{2} + {{\sqrt{\frac{p_{1}}{4M_{S}}}{K( {{X_{1}(x)} - {X_{1}( {\hat{x}}_{R} )}} )}}}^{2}} \rbrack}}} & \lbrack {{Formula}\mspace{14mu} 27} \rbrack \\{{m( {\lbrack {Y_{D_{1}},Y_{D_{2}}} \rbrack,{\overset{\_}{x}❘x},x_{R}} )} \approx \mspace{130mu}{{{{\sqrt{\frac{p_{1}}{M_{S}}}{G( {{X_{1}(x)} - {X_{1}( \overset{\_}{x} )}} )}} + N_{D_{1}}}}^{2} + \;\mspace{146mu}{\min\limits_{{\hat{x}}_{R} \in A^{L}}\lbrack {{{{\sqrt{\frac{p_{1}}{M_{S}}}{H( {{X_{2}(x)} = {X_{2}( \overset{\_}{x} )}} )}} + \mspace{149mu}{\sqrt{\frac{p_{3}}{M_{R}}}{F( {{X_{3}( x_{R} )} - {X_{3}( {\hat{x}}_{R} )}} )}} + N_{D_{2}}}}^{2} + {{\sqrt{\frac{p_{1}}{4M_{S}}}{K( {{X_{1}( \overset{\_}{x} )} - {X_{1}( {\hat{x}}_{R} )}} )}}}^{2}} \rbrack}}} & \lbrack {{Formula}\mspace{14mu} 28} \rbrack\end{matrix}$

Two cases of x_(R)=x and x_(R)≠x are considered to induce the averagepairwise error probability in the formula 24.

Firstly, for x_(R)=x, the matrices of the formulae 27 and 28 can berepresented as the following formulae 29 and 30.

$\begin{matrix}{ {{{m( {\lbrack {Y_{D_{1}},Y_{D_{2}}} \rbrack,x} }x},x_{R}} ) \approx {{N_{D_{1}}}^{2} + {N_{D_{2}}}^{2}}} & \lbrack {{Formula}\mspace{14mu} 29} \rbrack \\{{{ {{{m( {\lbrack {Y_{D_{1}},Y_{D_{2}}} \rbrack,x} }x},x_{R}} ){{{\sqrt{\frac{p_{1}}{M_{S}}}{G( {{X_{1}(x)} - {X_{1}( \overset{\Cup}{x} )}} )}} + N_{D_{1}}}}^{2}} + \mspace{14mu}{{{\sqrt{\frac{p_{1}}{M_{S}}}{H( {{X_{2}(x)} - {X_{2}( \overset{\Cup}{x} )}} )}\sqrt{\frac{p_{3}}{M_{R}}}{F( {{X_{3}(x)} - {X_{3}( {\overset{\Cup}{x}}_{R}^{\min} )}} )}} + N_{D_{2}}}}^{2} + {{\sqrt{\frac{p_{1}}{4M_{S}}}{K( {{X_{1}( \overset{\Cup}{x} )} - {X_{1}( {\overset{\bigwedge}{x}}_{R}^{\min} )}} )}}}^{2}}{{Here},{{\overset{\bigwedge}{x}}_{R}^{\min} = {{\min_{\overset{\bigwedge}{x}R^{\in A^{L}}}{{{\sqrt{\frac{p_{1}}{M_{S}}}{H( {{X_{2}(x)} - {X_{2}( \overset{\Cup}{x} )}} )}} + \mspace{175mu}{\sqrt{\frac{p_{3}}{M_{R}}}{F( {{X_{3}(x)} - {X_{3}( {\overset{\Cup}{x}}_{R} )}} )}} + N_{D_{2}}}}^{2}} + \mspace{310mu}{{{\sqrt{\frac{p_{1}}{4M_{S}}}{K( {{X_{1}( \overset{\Cup}{x} )} - {X_{1}( {\overset{\bigwedge}{x}}_{R} )}} )}}}^{2}.}}}}} & \lbrack {{Formula}\mspace{14mu} 30} \rbrack\end{matrix}$Then, for x_(R)=x, an expected value of a right-hand side of the formula24 can be represented as the following formula 31.

$\begin{matrix}{{{E\lbrack {P( {x->{\overset{\_}{x} x ){P_{SR}( x }x}} )} \rbrack} \leq {E( {Q\lbrack \frac{{{\sqrt{\frac{p_{1}}{M_{S}}}{G( {{X_{1}(x)} - {X_{1}( \overset{\_}{x} )}} )}}}^{2} + {{{\sqrt{\frac{p_{2}}{M_{s}}}{H( {{X_{2}(x)} - {X_{2}( \overset{\_}{x} )}} )}} + {\sqrt{\frac{p_{3}}{M_{R}}}{F( {{X_{3}(x)} - {X_{3}( {\overset{\hat{}\prime}{x}}_{R}^{\min} )}} )}}}}^{2} - {{\sqrt{\frac{p_{1}}{4M_{S}}}{K( {{X_{1}(x)} - {X_{1}( x_{R} )}} )}}}^{2}}{\sqrt{2{\sigma^{2}\lbrack {{{\sqrt{\frac{p_{1}}{M_{S}}}{G( {{X_{1}(x)} - {X_{1}( \overset{\_}{x} )}} )}}}^{2} + {{{\sqrt{\frac{p_{2}}{M_{s}}}{H( {{X_{2}(x)} - {X_{2}( \overset{\_}{x} )}} )}} + {\sqrt{\frac{p_{3}}{M_{R}}}{F( {{X_{3}( x_{R} )} - {X_{3}( {\hat{x}}_{R}^{\prime min} )}} )}}}}^{2}} \rbrack}}} )} \}} \leq {E\lbrack {Q( \sqrt{\frac{{{\sqrt{\frac{p_{1}}{M_{S}}}{G( {{X_{1}(x)} - {X_{1}( \overset{\_}{x} )}} )}}}^{2} + {{{\sqrt{\frac{p_{2}}{M_{s}}}{H( {{X_{2}(x)} - {X_{2}( \overset{\Cup}{x} )}} )}} + {\sqrt{\frac{p_{3}}{M_{R}}}{F( {{X_{3}(x)} - {X_{3}( {\hat{x}}_{R}^{\min} )}} )}}}}^{2} + {{\sqrt{\frac{p_{1}}{4M_{S}}}{K( {{X_{1}( \overset{\_}{x} )} - {X_{1}( {\hat{x}}_{R}^{\min} )}} )}}}^{2}}{2\sigma^{2}}} )} \rbrack}} = {{E\lbrack {\mathbb{e}}^{- \frac{{{\sqrt{\frac{p_{1}}{M_{S}}}{G{({{X_{1}{(x)}} - {X_{1}{(\overset{\_}{x})}}})}}}}^{2} + {{{\sqrt{\frac{p_{2}}{M_{S}}}{H{({{X_{2}{(x)}} - {X_{2}{(x)}}})}}} + {\sqrt{\frac{p_{3}}{M_{R}}}{F{({{X_{3}{(x)}} - {X_{3}{({\hat{x}}_{R}^{\min})}}})}}}}}^{2}}{4\sigma^{2}}} \rbrack}{E\lbrack {\mathbb{e}}^{- \frac{{{{\sqrt{\frac{p_{1}}{4M_{S}}}{K{(X)}}{(x)}} - {X_{1}{({\hat{x}}_{R}^{\min})}}}}^{2}}{4\sigma^{2}}} \rbrack}}} & \lbrack {{Formula}\mspace{11mu} 31} \rbrack\end{matrix}$

For x_(R)≠x, the matrices of the formulae 27 and 28 can be representedas the following formulae 32 and 33, respectively.

$\begin{matrix}{{m( {\lbrack {Y_{D_{1}},Y_{D_{2}}} \rbrack,{\overset{\Cup}{x}❘x},x_{R}} )} \leq {{N_{D_{1}}}^{2} + {N_{D_{2}}}^{2} + {{\sqrt{\frac{p_{1}}{4M_{S}}}{K( {{X_{1}(x)} - {X_{1}( x_{R} )}} )}}}^{2}}} & \lbrack {{Formula}\mspace{14mu} 32} \rbrack \\{{{m( {\lbrack {Y_{D_{1}},Y_{D_{2}}} \rbrack,{\overset{\Cup}{x}❘x},x_{R}} )} \geq \mspace{101mu}{{{{\sqrt{\frac{p_{1}}{M_{S}}}{G( {{X_{1}(x)} - {X_{1}( \overset{\Cup}{x} )}} )}} + N_{D_{1}}}}^{2} + \mspace{155mu}{{{\sqrt{\frac{p_{1}}{M_{S}}}{H( {{X_{2}(x)} - {X_{2}( \overset{\Cup}{x} )}} )}} + \mspace{225mu}{\sqrt{\frac{p_{3}}{M_{R}}}{F( {{X_{3}( x_{R} )} - {X_{3}( {\hat{x}}_{R}^{\prime\;\min} )}} )}} + N_{D_{2}}}}^{2}}}{{Here},\text{}{{\hat{x}}_{R}^{\prime\min} = {\min_{{\hat{x}}_{R} \in A^{L}}{{{{\sqrt{\frac{p_{1}}{M_{S}}}{H( {{X_{2}(x)} - {X_{2}( \overset{\Cup}{x} )}} )}} + \mspace{236mu}{\sqrt{\frac{p_{3}}{M_{R}}}{F( {{X_{3}( x_{R} )} - {X_{3}( {\hat{x}}_{R} )}} )}} + N_{D_{2}}}}^{2}.}}}}} & \lbrack {{Formula}\mspace{14mu} 33} \rbrack\end{matrix}$Then, an expected value of a right-hand side of the formula 24 can berepresented as the following formula 34.

$\begin{matrix}{{{E\lbrack {{P( {{x->\overset{\_}{x}}❘x_{R}} )}{P_{SR}( {x_{R}❘x} )}} \rbrack} \leq {E\lbrack {{Q( \frac{{{\sqrt{\frac{p_{1}}{M_{S}}}{G( {{X_{1}(x)} - {X_{1}( \overset{\_}{x} )}} )}}}^{2} + {{{\sqrt{\frac{p_{2}}{M_{S}}}{H( {{X_{2}(x)} - {X_{2}( \overset{\_}{x} )}} )}} + {\sqrt{\frac{p_{3}}{M_{R}}}{F( {{X_{3}(x)} - {X_{3}( {\hat{x}}_{R}^{\prime min} )}} )}}}}^{2} - {{\sqrt{\frac{p_{1}}{4M_{S}}}{K( {{X_{1}(x)} - {X_{1}( x_{R} )}} )}}}^{2}}{\sqrt{2{\sigma^{2}\lbrack {{{\sqrt{\frac{p_{1}}{M_{S}}}{G( {{X_{1}(x)} - {X_{1}( \hat{x} )}} )}}}^{2} + {{{\sqrt{\frac{p_{2}}{M_{S}}}{H( {{X_{2}(x)} - {X_{2}( \overset{\overset{\_}{\;}}{x} )}} )}} + {\sqrt{\frac{p_{3}}{M_{H}}}{F( {{X_{3}( x_{R} )} - {X_{3}( {\overset{\hat{}\prime}{x}}_{R}^{\min} )}} )}}}}^{2}} \rbrack}}} )}{Q( \sqrt{\frac{1}{2\sigma^{2}}{{\sqrt{\frac{p_{1}}{M_{S}}}{K( {{X_{1}(x)} - {X_{1}( x_{R} )}} )}}}^{2}} )}} \rbrack}}{{Here},\mspace{14mu}{{if}\mspace{14mu}{it}\mspace{14mu}{is}\mspace{14mu}{assumed}\mspace{14mu}{that}}}{l = {{{\sqrt{\frac{p_{1}}{M_{S}}}{G( {{X_{1}(x)} - {X_{1}( \overset{\Cup}{x} )}} )}}}^{2} + {{{\sqrt{\frac{p_{2}}{M_{S}}}{H( {{X_{2}(x)} - {X_{2}( \overset{\Cup}{x} )}} )}} + {\sqrt{\frac{p_{3}}{M_{R}}}{F( {{X_{3}( x_{R} )} - {X_{3}( {\hat{x}}_{R}^{\prime min} )}} )}}}}^{2}}}} & \lbrack {{Formula}\mspace{11mu} 34} \rbrack\end{matrix}$and

${q = {{\sqrt{\frac{p_{1}}{4M_{S}}}{K( {{X_{1}(x)} - {X_{1}( x_{R} )}} )}}}^{2}},$the formula 34 can be represented as the following formula 35.

$\begin{matrix}{{E\lbrack {{P( {{x->\overset{\Cup}{x}}❘x_{R}} )}{P_{SR}( {x_{R}❘x} )}} \rbrack} \leq {E\lbrack {{Q( \frac{l - q}{\sqrt{2\sigma^{2}l}} )}{Q( \sqrt{\frac{2q}{\sigma^{2}}} )}} \rbrack}} & \lbrack {{Formula}\mspace{14mu} 35} \rbrack\end{matrix}$

Here, if l>q, a Q function will have an upper limit value of

${\mathbb{e}}^{- \frac{{({l - q})}^{2}}{4l\;\sigma^{2}}}.$Otherwise, the upper limit value is 1. An expected value of the formula35 has an upper limit value, represented as the following formula 36.

$\begin{matrix}{{{E\lbrack {{Q( \frac{l - q}{\sqrt{2\sigma^{2}l}} )}{Q( \sqrt{\frac{2q}{\sigma^{2}}} )}} \rbrack} \leq {{\int_{0}^{\infty}{\int_{l}^{\infty}{{\mathbb{e}}^{- \frac{q}{\sigma^{2}}}f_{q}f_{l}{\mathbb{d}q}\mspace{14mu}{\mathbb{d}l}}}} + {\int_{0}^{\infty}{\int_{0}^{l}{{\mathbb{e}}^{- \frac{q}{\sigma^{2}}}{\mathbb{e}}^{- \frac{{({l - q})}^{2}}{4l\;\sigma^{2}}}f_{q}f_{l}\mspace{11mu}{\mathbb{d}q}\mspace{14mu}\mathbb{d}}}}} \leq {{\int_{0}^{\infty}{\int_{l}^{\infty}{{\mathbb{e}}^{- \frac{l + {2q}}{4\sigma^{2}}}f_{q}f_{l}\mspace{11mu}{\mathbb{d}q}\mspace{14mu}{\mathbb{d}l}}}} + {\int_{0}^{\infty}{\int_{0}^{l}{{\mathbb{e}}^{- \frac{l + {2q}}{4\sigma^{2}}}f_{q}f_{l}{\mathbb{d}q}\mspace{14mu}{\mathbb{d}l}}}}}} = {E\lbrack {\mathbb{e}}^{- \frac{l + {2q}}{4\sigma^{2}}} \rbrack}} & \lbrack {{Formula}\mspace{14mu} 36} \rbrack\end{matrix}$

Accordingly, the formula 34 can be represented as the following formula37.

$\begin{matrix}{{E\lbrack {{P( {{x->\overset{\Cup}{x}}❘x_{R}} )}{P_{SR}( {x_{R}❘x} )}} \rbrack} \leq {E\lbrack {\mathbb{e}}^{- \frac{{{\sqrt{\frac{p_{1}}{M_{S}}}{G{({{X_{1}{(x)}} - {X_{1}{(\overset{\_}{x})}}})}}}}^{2} + {{{\sqrt{\frac{p_{2}}{M_{S}}}{H{({{X_{2}{(x)}} - {X_{2}{(\overset{\_}{x})}}})}}} + {\sqrt{\frac{p_{3}}{M_{R}}}{F{({{X_{3}{(x)}} - {X_{3}{({\overset{\hat{}\prime}{x}}_{R}^{\min})}}})}}}}}^{2} + {{\sqrt{\frac{p_{1}}{4M_{S}}}{K{({{X_{1}{(\overset{\_}{x})}} - {X_{1}{(x_{R})}}})}}}}^{2}}{4\sigma^{2}}} \rbrack}} & \lbrack {{Formula}\mspace{14mu} 37} \rbrack\end{matrix}$

Moreover, from the formulae 31 and 37, the average pairwise errorprobability in the formula 24 has an upper limit value, represented asthe following formula 38.

$\begin{matrix}{{E\lbrack {P( {x->\overset{\Cup}{x}} )} \rbrack} \leq {{{E\lbrack e^{- \frac{{{\sqrt{\frac{p_{1}}{M_{S}}}{G{({{X_{1}{(x)}} - {X_{1}{(\overset{\_}{x})}}})}}}}^{2} + {{{\sqrt{\frac{p_{2}}{M_{S}}}{H{({{X_{2}{(x)}} - {X_{2}{(\overset{\_}{x})}}})}}} + {\sqrt{\frac{p_{3}}{M_{R}}}{F{({{X_{3}{(x)}} - {X_{3}{({\hat{x}}_{R}^{\min})}}})}}}}}^{2}}{4\sigma^{2}}} \rbrack}{E\lbrack {\mathbb{e}}^{- \frac{{{\sqrt{\frac{p_{1}}{4M_{S}}}{K{({{X_{1}{(x)}} - {X_{1}{({\hat{x}}_{R}^{\min})}}})}}}}^{2}}{4\sigma^{2}}} \rbrack}} + {\sum\limits_{x_{R} \neq x}{{E\lbrack {\mathbb{e}}^{- \frac{{{\sqrt{\frac{p_{1}}{M_{S}}}{G{({{X_{1}{(x)}} - {X_{1}{(\overset{\_}{x})}}})}}}}^{2} + {{{\sqrt{\frac{p_{2}}{M_{S}}}{H{({{X_{2}{(x)}} - {X_{2}{(\overset{\_}{x})}}})}}} + {\sqrt{\frac{p_{3}}{M_{R}}}{F{({{X_{3}{(x)}} - {X_{3}{({\hat{x}}_{R}^{\min})}}})}}}}}^{2}}{4\sigma^{2}}} \rbrack}{E\lbrack {\mathbb{e}}^{- \frac{{{\sqrt{\frac{p_{1}}{4M_{S}}}{K{({{X_{1}{(x)}} - {X_{1}{(x_{R})}}})}}}}^{2}}{4\sigma^{2}}} \rbrack}}}}} & \lbrack {{Formula}\mspace{11mu} 38} \rbrack\end{matrix}$

In this case, since {circumflex over (x)}_(R) ^(min) and {circumflexover (x)}′_(R) ^(min) are changed according to the change of K, G, H,and F (i.e., {circumflex over (x)}_(R) ^(min) and {circumflex over(x)}′_(R) ^(min) are x, {circumflex over (x)}_(R), x_(R), or othersymbol). It is typically impossible to calculate an expected value ofthe formula 38. However, for the NDF-SAS protocol, the formula 38 can berepresented as the following formula 39.

$\begin{matrix}{{E\lbrack {P( {x->\overset{\Cup}{x}} )} \rbrack} \leq {{E\lbrack {\mathbb{e}}^{- \frac{{{\sqrt{\frac{p_{1}}{M_{S}}}{G{({{X_{1}{(x)}} - {X_{1}{(\overset{\_}{x})}}})}}}}^{2}}{4\sigma^{2}}} \rbrack}\{ {{{E\lbrack {\mathbb{e}}^{- \frac{{{{\sqrt{\frac{p_{2}}{M_{S}}}{H{({{X_{2}{(x)}} - {X_{2}{(\overset{\_}{x})}}})}}} + {\sqrt{\frac{p_{3}}{M_{R}}}{F{({{X_{3}{(x)}} - {X_{3}{({\hat{x}}_{R}^{\min})}}})}}}}}^{2}}{4\sigma^{2}}} \rbrack}{E\lbrack {\mathbb{e}}^{- \frac{{{\sqrt{\frac{p_{1}}{4M_{S}}}{K{({{X_{1}{(x)}} - {X_{1}{({\hat{x}}_{R}^{\min})}}})}}}}^{2}}{4\sigma^{2}}} \rbrack}} + {\sum\limits_{x_{R} \neq x}{{E\lbrack {\mathbb{e}}^{- \frac{{{{\sqrt{\frac{p_{2}}{M_{S}}}{H{({{X_{2}{(x)}} - {X_{2}{(\overset{\_}{x})}}})}}} + {\sqrt{\frac{p_{3}}{M_{R}}}{F{({{X_{3}{(x)}} - {X_{3}{({\overset{\hat{}\prime}{x}}_{R}^{\min})}}})}}}}}^{2}}{4\sigma^{2}}} \rbrack}{E\lbrack {\mathbb{e}}^{- \frac{{{\sqrt{\frac{p_{1}}{2M_{S}}}{K{({{X_{2}{(x)}} - {X_{1}{(x_{H})}}})}}}}^{2}}{4\sigma^{2}}} \rbrack}}}} \}}} & \lbrack {{Formula}\mspace{14mu} 39} \rbrack\end{matrix}$

For the NDF protocol, the formula 38 can be represented as the followingformula 40.

$\begin{matrix}{{E\lbrack {P( {x->\overset{\Cup}{x}} )} \rbrack} \leq {{{E\lbrack {\mathbb{e}}^{- \frac{{{\sqrt{\frac{p_{1}}{M_{S}}}{G{({{X_{1}{(x)}} - {X_{1}{(\overset{\_}{x})}}})}}}}^{2} + {{{\sqrt{\frac{p_{2}}{M_{S}}}{H{({{X_{2}{(x)}} - {X_{2}{(\overset{\_}{x})}}})}}} + {\sqrt{\frac{p_{3}}{M_{R}}}{F{({{X_{3}{(x)}} - {X_{3}{({\overset{\hat{}\prime}{x}}_{R}^{\min})}}})}}}}}^{2}}{4\sigma^{2}}} \rbrack}{E\lbrack {\mathbb{e}}^{- \frac{{{\sqrt{\frac{p_{1}}{4M_{S}}}{K{({{X_{1}{(x)}} - {X_{1}{({\hat{x}}_{R}^{\min})}}})}}}}^{2}}{4\sigma^{2}}} \rbrack}} + {\sum\limits_{x_{R} \neq x}{{E\lbrack {\mathbb{e}}^{- \frac{{{\sqrt{\frac{p_{1}}{M_{S}}}{G{({{X_{1}{(x)}} - {X_{1}{(\overset{\_}{x})}}})}}}}^{2} + {{{\sqrt{\frac{p_{2}}{M_{S}}}{H{({{X_{2}{(x)}} - {X_{2}{(\overset{\_}{x})}}})}}} + {\sqrt{\frac{p_{3}}{M_{R}}}{F{({{X_{3}{(x_{R})}} - {X_{3}{({\hat{x}}_{R}^{\prime min})}}})}}}}}^{2}}{4\sigma^{2}}} \rbrack}{E\lbrack {\mathbb{e}}^{- \frac{{\begin{matrix}{\sqrt{\frac{p_{1}}{4M_{S}}}{K({{X_{1}{(x)}} -}}} \\{{X_{1}{(x_{R})}})}\end{matrix}}^{2}}{4\sigma^{2}}} \rbrack}}}}} & \lbrack {{Formula}\mspace{14mu} 40} \rbrack\end{matrix}$

A summation of right-hand side expected values in the parenthesis of theformula 39 has the same diversity order as that of the right-hand sideof the following formula 40. Accordingly, the diversity order of theNDF-SAS protocol is greater than that of the NDF protocol by M_(S)M_(D).

Next, in the case of the error SR channel, for M_(S)=M_(R)=1 andM_(S)=M_(R)=2, the average bit error probabilities of the NDF-SASprotocol and the NDF protocol is compared with those of the Alamoutischeme and the CISTBC scheme. Here, conditions of a simulation areidentical to that of the aforementioned error-free SR channel. In FIG.7, when σ² _(SR)=10, σ² _(SR)=1, and σ² _(SR)=0.1, forM_(S)=M_(R)=M_(D)=1, the average bit error probability of the Alamoutischeme for the NDF-SAS protocol and the NDF protocol is compared withthat of the error-free SR channel (‘NER’). In FIG. 8, when σ² _(SR)=10,σ² _(SR)=1, and σ² _(SR)=0.1 for M_(S)=M_(R)=2 and M_(D)=1, the averagebit error probability of the CISTBC scheme for the NDF-SAS protocol andthe NDF protocol is compared with that of the error-free SR channel.From FIGS. 7 and 8, when σ² _(SR)≧1, for M_(D)=1, the average bit errorprobability of the error SR channel is similar to that of the error-freeSR channel. When σ² _(SR)=0.1, the average bit error probability isdifferent from that of the error-free SR channel. However, the averagebit error probability has the same diversity for the NDF-SAS protocoland the NDF protocol. In other words, the diversity of the NDF-SASprotocol is improved by M_(S)M_(D)

As described above, the present invention provides a source antennaswitching scheme for a non-orthogonal decode-and-forward protocol thatcan acquire a greater diversity than the conventional NDF protocol. Inother words, the present invention provides a source antenna switchingscheme for a non-orthogonal decode-and-forward protocol that canincrease a diversity order by adding a reasonable priced antenna insteadof expensive hardware such as an RF chain when there are a plurality ofantenna in the RF chain. Further, the present invention provides asource antenna switching scheme for a non-orthogonal decode-and-forwardprotocol that can have no loss of encoding rate as compared with aconventional protocol that has a relay node without switching a sourceantenna even if a source node antenna is switched. In additional, thepresent invention provides a source antenna switching scheme for anon-orthogonal decode-and-forward protocol that can easily decode asignal transmitted from a source node to a relay node by using a near MLdecoding method instead of an ML decoding method.

While the invention has been shown and described with respect to theembodiments, it will be understood by those skilled in the art thatvarious changes and modification may be made without departing from thescope of the invention as defined in the following claims.

What is claimed is:
 1. A source antenna switching method for anon-orthogonal protocol that transmits a signal of a source node throughat least one RF chain having two transmitting antennas, the sourceantenna switching method comprising: selecting one of the twotransmitting antennas of the at least one RF chain and allowing thesource node to transmit the signal to a relay node and a destinationnode by using the selected antenna utilizing a total of M_(S)quantities; and selecting other of the two transmitting antennas andallowing the other antenna in the M_(S) quantities to cooperate with anantenna of the relay node to transmit a signal to destination node,wherein the M_(S) quantities are number of RF chains of the source node;wherein the selecting of the other antenna and allowing the otherantenna in the M_(S) quantities to cooperate with the antenna of therelay node comprises: allowing the source node and the relay node togenerate a distributed space-time code and transmit the generateddistributed space-time code to the destination node; decoding oramplifying the signal received by the relay node; and generating aspace-time code by using the decoded or amplified signal to betransmitted by the source node by use of the other antenna in the M_(S)quantities and transmitting the generated space-time code to thedestination node.
 2. The source antenna switching method of claim 1,wherein the selecting of the other antenna and allowing the otherantenna in the M_(S) quantities to cooperate with an antenna of therelay node comprises: decoding the signal by the relay node receivedfrom the source node; generating a space-time code by using the decodedsignal to be transmitted by the source node by use of the other antennain the M_(S) quantities and transmitting the generated space-time codeto the destination node; and decoding by using a near ML decoding methoda signal that is received through the selecting of one of the twotransmitting antennas of the at least one RF chain and allowing thesource node to transmit the signal to the relay node and the destinationnode by using the selected one antenna in the total of M_(S) quantitiesand the selecting of the other of the two antennas and allowing theother antenna in the M_(S) quantities to cooperate with an antenna(s) ofthe relay node to transmit a signal to destination node.
 3. The sourceantenna switching method of claim 2, wherein the near ML decoding methodis performed by a following formula$\overset{.}{x} = {\arg\mspace{14mu}{\min\limits_{x_{\in}A^{L}}\{ {{{Y_{D_{1}} - {\sqrt{\frac{p_{1}}{M_{S}}}{{GX}_{1}(x)}}}}^{2} + \mspace{11mu}{\min\limits_{{\hat{x}}_{R} \in A^{L}}{{Y_{D_{2}} - {\sqrt{\frac{p_{20}}{M_{S}}}{{HX}_{2}(x)}} - {\sqrt{\frac{p_{3}}{M_{R}}}{{FX}_{3}( {\hat{x}}_{R} )}}}}^{2}} - \lbrack {\sigma^{2}\ln\mspace{14mu}{P_{SR}( {x->{\hat{x}}_{R}} )}} \rbrack} \}}}$where G is a channel coefficient matrix of a channel between the sourcenode and the destination node in the selecting of one of the twoantennas of the at least one RF chain and allowing the source node totransmit the signal to the relay node and the destination node by usingthe selected one antenna in the M_(S) quantities, H is a channelcoefficient matrix of a channel between the source node and thedestination node in the selecting of the other antenna and allowing theother antenna in the M_(S) quantities to cooperate with the antenna ofthe relay node; F is a channel coefficient matrix of a channel betweenthe relay node and the destination node in the selecting of the otherantenna and allowing the other antenna in the M_(S) quantities tocooperate with the antenna of the relay node, x is L data symbolstransmitted from the source node through the selecting of one of the twoantennas of the least one RF chain and allowing the source node totransmit the signal to the relay node and the destination node by usingthe selected one antenna in the M_(S) quantities and the selecting ofthe other antenna and allowing the other antenna in the M_(S) quantitiesto cooperate with the antenna of the relay node, M_(R) is a number oftransmitting and receiving antennas of the relay node, X₁(x) is a codeof M_(S)×T₁ of the L data symbols, X₂(x) is a code of M_(S)×T₂ of the Ldata symbols, X₃({circumflex over (x)}_(R)) is a code of M_(R)×T₂ of theL data symbols, the p₁ is a power of a signal transmitted from thesource node in a first operation, p₂ is a power supplied from the sourcenode in a second operation, p₃ is a power supplied from the relay nodein the second operation, σ² is a power of noise in the relay node andthe destination node, P_(sr) is a pairwise error probability, S is thesource node, is the relay node, and D is the destination node.
 4. Thesource antenna switching method of claim 1, wherein the two transmittingantennas of the at least one RF chain operate independently from eachother.